ON FLOW POLYTOPES, ORDER POLYTOPES, AND CERTAIN FACES OF THE ALTERNATING SIGN MATRIX POLYTOPE

KAROLA MESZ´ AROS,´ ALEJANDRO H. MORALES, JESSICA STRIKER

Abstract. In this paper we study an alternating sign matrix analogue of the Chan-Robbins-Yuen polytope, which we call the ASM-CRY polytope. We show that this polytope has Catalan many vertices and its volume is equal to the number of standard Young tableaux of staircase shape; we also determine its Ehrhart polynomial. We achieve the previous by proving that the members of a family of faces of the alternating sign matrix polytope which includes ASM-CRY are both order and flow polytopes. Inspired by the above results, we relate three established triangulations of order and flow polytopes, namely Stanley’s triangulation of order polytopes, the Postnikov-Stanley triangulation of flow polytopes and the Danilov-Karzanov-Koshevoy triangulation of flow polytopes. We show that when a graph G is a planar graph, in which case the flow polytope FG is also an order polytope, Stanley’s triangulation of this order polytope is one of the Danilov-Karzanov-Koshevoy triangulations of FG. Moreover, for a general graph G we show that the set of Danilov-Karzanov- Koshevoy triangulations of FG equal the set of framed Postnikov-Stanley triangulations of FG. We also describe explicit bijections between the combinatorial objects labeling the simplices in the above triangulations.

Contents 1. Introduction 1 2. Faces of the Birkhoff and alternating sign matrix polytopes 3 3. Flow and order polytopes 5 4. Proofs of Theorem 3.11 and Theorem 3.14 9 5. (n) and the family of polytopes (ASM) 12 6.ASMCRY Triangulations of flow polytopes of planar graphsF 17 7. Triangulations of flow polytopes of general graphs 21 Acknowledgments 28 References 28

1. Introduction

arXiv:1510.03357v2 [math.CO] 15 Jan 2019 In this paper we study a family of faces of the alternating sign matrix polytope inspired by an intriguing face of the Birkhoff polytope: the Chan-Robbins-Yuen (CRY) polytope [8]. We call these faces the ASM-CRY family of polytopes. Interest in the CRY polytope centers around its volume formula as a product of consecutive Catalan numbers; this has been proved [31] via an identity equivalent to the Selberg integral, but the problem of finding a combinatorial proof remains open. We prove that the polytopes in the ASM-CRY family are order polytopes and use

KM was partially supported by a National Science Foundation Grant (DMS 1501059) as well as by a von Neumann Fellowship at the IAS funded by the Fund for Mathematics and the Friends of the Institute for Advanced Study. AHM was partially supported by a CRM-ISM Postdoctoral Fellowship and an AMS-Simons travel grant. JS was partially supported by a National Security Agency Grant (H98230-15-1-0041), the North Dakota EPSCoR National Science Foundation Grant (IIA-1355466), the NDSU Advance FORWARD program sponsored by National Science Foundation grant (HRD-0811239), and a grant from the Simons Foundation/SFARI (527204, JS). 1 Stanley’s theory of order polytopes [26] to give a combinatorial proof of formulas for their volumes and Ehrhart polynomials. We also show that these polytopes, and all order polytopes of strongly planar posets, are flow polytopes. of planar graphs (Theorem 3.14). The converse of this statement is due to Postnikov [21] (private communication) and we here include a proof (Theorem 3.11). These observations bring us to the general question of relating the different known triangulations of flow and order polytopes. We show that when G is a planar graph, in which case the flow polytope of G is also an order polytope, then Stanley’s canonical triangulation of this order polytope [26] is one of the Danilov-Karzanov-Koshevoy triangulations of the flow polytope of G [9], a statement first observed by Postnikov [21]. Moreover, for general G we show that the set of Danilov-Karzanov-Koshevoy triangulations of the flow polytope of G equals the set of framed Postnikov-Stanley triangulations of the flow polytope of G [21, 25]. We also describe explicit bijections between the combinatorial objects labeling the simplices in the above triangulations, answering a question posed by Postnikov [21]. We highlight the main results of the paper in the following theorems. While we define some of the notation here, some only appears in later sections to which we give pointers after the relevant statements. In Definition 5.1, we define the ASM-CRY family (ASM)(n) of polytopes (n) indexed by F Pλ partitions λ δn where δn := (n 1, n 2,..., 1). In Theorem 5.3, we prove that the polytopes in this family⊆ are faces of the alternating− − sign matrix polytope (n) defined in [5, 29]. In the case A when λ = ∅ we obtain an analogue of the Chan-Robbins-Yuen (CRY) polytope, which we call the ASM-CRY polytope, denoted by (n). This polytope contains the CRY polytope. Our main theorem about the family ofASMCRY polytopes (ASM)(n) is the following. For the necessary definitions, see Sections 3.3 and 5. F Theorem 1.1. The polytopes in the family (ASM)(n) are integrally equivalent to flow and order F polytopes. In particular, λ(n) is integrally equivalent to the order polytope of the poset (δn λ)∗ and the flow polytope P . \ FG(δn\λ)∗ By Stanley’s theory of order polytopes [26] it follows that the volume of the polytope Pλ(n) for any P (n) (ASM)(n) is given by the number of linear extensions of the poset (δ λ) (which λ ∈ F n \ ∗ equals the number of standard Young tableaux of skew shape δn/λ), and its Ehrhart polynomial in the variable t is given by the order polynomial of the poset (δn λ)∗. See Corollary 5.7 for the general statement. We give the application to (n) in the\ corollary below. For further examples of polytopes in (ASM)(n), see FigureASMCRY 9. F Corollary 1.2. (n) is integrally equivalent to the order polytope of the poset δ . Thus, ASMCRY n∗ (n) has Cat(n) = 1 2n vertices, its normalized volume is given by ASMCRY n+1 n vol(
(n)) = #SYT (δ ), ASMCRY n and its Ehrhart polynomial is 2t + i + j 1 (1.1) L (n)(t) = Ωδ∗ (t + 1) = − . ASMCRY n i + j 1 1 i

2. Faces of the Birkhoff and alternating sign matrix polytopes In this section, we explain the motivation for our study of certain faces of the alternating sign matrix polytope. We review some standard facts of lattice point enumeration of integral polytopes [4],[27, 4.6]. Given an integral polytope , we denote by relvol( ) the volume of relative to its lattice and§ by L (t) the Ehrhart functionP that counts the numberP of lattice pointsP of the dilated polytope t . AP well known result of Ehrhart [11] states that if is integral, then L (t) is a polynomial·P of degree dim( ) with leading coefficient relvol( ) (seeP [4, Cor. 3.16]). TheP quantity dim( )! relvol( ) is an integerP (see [4, Cor. 3.17]) called theP normalized volume that we denote by vol(P ·). P P We say that two integral polytopes in Rd and in Rm are integrally equivalent, which int P Q we denote by , if there is an affine transformation ϕ : Rd Rm whose restriction to P ≡ Q → P is a bijection ϕ : that preserves the lattice, i.e. ϕ is a bijection between Zd aff( ) and P → Q ∩ P Zm aff( ) where aff( ) denotes the affine span. The map ϕ is then an integral equivalence. Note∩ thatQ integrally equivalent· polytopes have the same Ehrhart polynomials and therefore they have the same volume. We remark that isomorphism and unimodular equivalence are other terms sometimes used in the literature for what we will refer to as integral equivalence. Next we define the Birkhoff and Chan-Robbins-Yuen polytopes; we then define the alternating sign matrix counterparts. 3 Definition 2.1. The Birkhoff polytope, (n), is defined as B n n2 (n) := (bij) R bij 0, bij = 1, bij = 1 . B i,j=1 ∈ | ≥ n Xi Xj o Matrices in (n) are called doubly-stochastic matrices. A well-known theorem of Birkhoff [6] and von NeumannB [30] states that (n), as defined above, equals the convex hull of the n n permutation matrices. Note that (n)B has n2 facets and dimension (n 1)2, its vertices are× the permutation matrices, and its volumeB has been calculated up to n = 10− by Beck and Pixton [3]. De Loera, Liu and Yoshida [10] gave a closed summation formula for the volume of (n), which, while of interest on its own right, does not lend itself to easy computation. Shortly after,B Canfield and McKay [7] gave an asymptotic formula for the volume. A special face of the Birkhoff polytope, first studied by Chan-Robbins-Yuen [8], is as follows. Definition 2.2. The Chan-Robbins-Yuen polytope, (n), is defined as CRY (n) := (b )n (n) b = 0 for i j 2 . CRY ij i,j=1 ∈ B | ij − ≥ n n 1 (n) has dimension 2 and 2 − vertices. This polytope was introduced by Chan-Robbins- YuenCRY [8] and in [31] Zeilberger calculated its normalized volume as the following product of Catalan numbers.
Theorem 2.3 (Zeilberger [31]). n 2 − vol( (n)) = Cat(i) CRY Yi=1 1 2i where Cat(i) = i+1 i . The proof in [31] used
a relation (see Theorem 3.4) expressing the volume as a value of the Kostant partition function (see Definition 3.5) and a reformulation of the Morris constant term identity [20] to calculate this value. No combinatorial proof is known. Next we give an analogue of the Birkhoff polytope in terms of alternating sign matrices. Recall that alternating sign matrices (ASMs) [17] are square matrices with the following properties: entries 0, 1, 1 , • the entries∈ { in each− } row/column sum to 1, and • the nonzero entries along each row/column alternate in sign. • The ASMs with no negative entries are the permutation matrices. See Figure 1 for an example.

1 0 0 1 0 0 0 1 0 0 1 0 0 1 0 0 0 1 0 0 1 0 1 0 0 0 1 1 0 0 1 1 1 0 0 1 1 0 0 0 1 0  0 0 1   0 1 0   0 0 1   0− 1 0   1 0 0   0 1 0   1 0 0                Figure 1. All the 3 3 alternating sign matrices. ×

Definition 2.4 (Behrend-Knight [5], Striker [29]). The alternating sign matrix polytope, (n), is defined as follows: A i0 j0 n n n n2 (n) := (aij) R 0 aij 1, 0 aij 1, aij = 1, aij = 1 , A i,j=1 ∈ | ≤ ≤ ≤ ≤ n Xi=1 Xj=1 Xi=1 Xj=1 o where we have the first sum for any 1 i0, j n, the second sum for any 1 j0, i n, the third sum for any 1 j n, and the fourth≤ sum for≤ any 1 i n. ≤ ≤ ≤ ≤ 4 ≤ ≤ 1 0 0 1 0 0 0 1 0 0 1 0 0 1 0 0 1 0 1 0 0 1 0 0 0 0 1 0 0 1 0 0 1 0 0 1

0 0 1 0 0 1 1 0 0 1 0 0 .4 .2 .1 .3 .3 .4 .1 .2 0 1 0 0 1 0 .6 .1 0 .3 .7 -.2 -.1 .4 1 0 0 0 1 0 1 0 0 0 .7 .2 .1 0 .8 .1 .1 0 0 1 1 1 1 0 0 1 (3) 0 1 0 0 1 0 (3) 0 1 0 0 0 .7 .3 0 0 .9 .1 CRY ASMCRY (a) (b) (c) (d)

Figure 2. (a) The polytope (3) in R3, (b) the polytope (3) in R3, (c) a doubly-stochastic matrixCRY in (4), (d) a matrix in ASMCRY(4). CRY ASMCRY

Behrend and Knight [5], and independently Striker [29], defined (n). The alternating sign matrix polytope can be seen as an analogue of the Birkhoff polytope, sinceA the former is the convex hull of all alternating sign matrices (which include all permutation matrices) while the latter is the convex hull of all permutation matrices. The polytope (n) has 4((n 2)2 + 1) facets (for n 3) [29], its dimension is (n 1)2, and its vertices are the nA n alternating− sign matrices [5, 29]. The≥ Ehrhart polynomial has been− calculated up to n = 5 [5]. Its× normalized volume for n = 1,..., 5 is calculated to be 1, 1, 4, 1376, 201675688, and no asymptotic formula for its volume is known. In analogy with (n), we study a special face of the ASM polytope we call the ASM-CRY polytope (and show,CRY in Theorem 5.3, it is indeed a face of (n)). A Definition 2.5. The ASM-CRY polytope is defined as follows. (n) := (a )n (n) a = 0 for i j 2 . ASMCRY ij i,j=1 ∈ A | ij − ≥ Since the (n) polytope has a nice product formula for its normalized volume, it is then natural to wonderCRY if the volume of the alternating sign matrix analogue of (n), which we denote by (n), is similarly nice. In Theorem 1.1 and Corollary 1.2, weCRY show that (n) is bothASMCRY a flow and order polytope, and using the theory established for the latter, we giveASMCRY the volume formula and the Ehrhart polynomial of (n). Just like in the (n) case, all formulas obtained are combinatorial. Unlike in theASMCRY(n) case, all the proofs involvedCRY are combinatorial. In Theorem 1.1, we extend these results toCRY a family of faces (ASM)(n) of the ASM polytope, of which (n) is a member; see Section 5. F ASMCRY 3. Flow and order polytopes In order to state and prove Theorem 1.1 in Section 5, we need to discuss flow and order polytopes. In Section 3.1, we define flow and order polytopes and also explain how to see (n) as the flow polytope of the complete graph. In Sections 3.2 and 3.3, we state in TheoremsCRY 3.11 and 3.14 that the flow polytope of a planar graph is the order polytope of a related poset, and vice versa. We give the proofs of these theorems in Section 4.

3.1. Background and definitions. Let G be a connected graph on the vertex set [n] := 1, 2, . . . , n with edges directed from the smaller to larger vertex. Denote by in(e) the smaller (initial){ vertex } of edge e and fin(e) the bigger (final) vertex of edge e. 5 n 1 Definition 3.1. Given a vector a = (a1, a2, . . . , an 1, i=1− ai) with ai Z 0, a flow fl on G − − ∈ ≥ with netflow a is a function fl : E(G) R 0 such that for i = 1, 2, . . . , n 1 → ≥ P − fl(e) fl(e) = a − i e E,in(e)=i e E,fin(e)=i ∈ X ∈ X and n 1 − fl(e) = ai. e E,fin(e)=n i=1 ∈ X X The flow polytope (a) associated to the graph G and netflow vector a is the set of all flows FG int fl : E(G) R 0 on G with netflow a. We denote the set of integer flows of G(a) by G (a). → ≥ F F Definition 3.2. A flow fl of size one on G is a flow on G with netflow (1, 0,..., 0, 1). That is − fl(e) = fl(e) = 1, e E,in(e)=1 e E,fin(e)=i ∈ X ∈ X and for 2 i n 1 ≤ ≤ − fl(e) = fl(e). e E,fin(e)=i e E,in(e)=i ∈ X ∈ X The flow polytope G associated to the graph G is the set of all flows fl : E(G) R 0 of size F → ≥ one on G.

We assume that in our flow polytopes G each vertex v 2, 3, . . . , n 1 in G has both incoming and outgoing edges. Note that this restrictionF is not a serious∈ { one. If there− } is a vertex v [2, n 1] with only incoming or outgoing edges, then in the flow on all these edges must be∈ zero,− and FG thus, up to removing such vertices, any flow polytope G is integrally equivalent to a flow polytope defined as above. F #E(G) The polytope G is a convex polytope in the Euclidean space R and its dimension is dim( ) = #E(GF) #V (G) + 1 (e.g. see [1]). The vertices of are characterized as follows. FG − FG Proposition 3.3 ([13, Cor. 3.1]). Let G be a connected graph with vertices [n] with edges oriented from smaller to bigger vertices. Then the vertices of G are the unit flows on maximal directed paths or routes from the source 1 to the sink n. F Figure 3 shows the equations of and explains why this polytope is integrally equivalent to FK5 (4). The same correspondence shows that Kn+1 and (n) coincide. The following theorem CRYconnects volumes of flow polytopes and KostantF partitionCRY functions. Theorem 3.4 (Postnikov-Stanley [21, 25], Baldoni-Vergne [1]). For a loopless graph G on the vertex set 1, 2 . . . , n , with d = indeg (G) 1, { } i i − n 1 − vol ( G) = KG(0, d2, . . . , dn 1, di), F − − Xi=2 where KG(a) is the Kostant partition function, indegi(G) denotes the indegree of vertex i in G and vol is normalized volume. Recall the definition of the Kostant partition function.

Definition 3.5. The Kostant partition function KG(v) is the number of ways to write the vector v as a nonnegative linear combination of the positive type An 1 roots corresponding to the − edges of G, without regard to order. The edge (i, j), i < j, of G corresponds to the vector ei ej, th n − where ei is the i standard basis vector in R . 6 K d 5 a b c d c g 1= a + b + c + d 0= e + f + g a e f g b f i − a e h j 0= h + i b e 0 h i − − 0= j c f h 0 0 j 1 0 0 0 -1 − − − Figure 3. Graph K5 with edges directed from smaller to bigger vertex. The flow variables on the edges are a, b, c, d, e, f, g, h, i, j, the net flows in the vertices are 1, 0, 0, 0, 1. The equations defining the flow polytope corresponding to K5 are in the middle.− Note that these same equations define (4) as can be seen from the matrix on the left, where we denoted by entries thatCRY are determined by the variables a, b, . . . , j. •

It is easy to see by definition that the number of integer flows on G with netflow a, that is, the int size of G (a) or number of integer points in the flow polytope G(a), equals KG(a). In particular, the EhrhartF polynomial of in variable t is equal to K (t, 0,...,F 0, t). FG G − Now we are ready to define order polytopes and relate them to flow polytopes.

Definition 3.6 (Stanley [26]). The order polytope, (P ), of a poset P with elements t1, t2, . . . , tn n O { } is the set of points (x1, x2, . . . , xn) in R with 0 xi 1 and if ti P tj then xi xj. We identify ≤ ≤ ≤ ≤ each point (x1, x2, . . . , xn) of (P ) with the function f : P R with f(ti) = xi. O → In our proofs, we will often use the polytope ˆ(P ), which is integrally equivalent to (P ) [26, Sec. 1]: O O Definition 3.7 (Stanley [26]). Let P be the poset obtained from P by adjoining a minimum element 0ˆ and a maximum element 1.ˆ Define a polytope (P ) to be the set of functions g : P R O → satisfying g(0)ˆ = 0, g(1)ˆ = 1, and g(x)b g(y) if x y in P . ≤ ≤ b b Lemma 3.8 (Stanley [26]). The map ν : (P ) (P ) given by (g(x)) (g(x)) is an b x Pb x P integral equivalence. O → O ∈ 7→ ∈ b In general, computing or finding a combinatorial interpretation for the volume of a polytope is a hard problem. Order polytopes are an especially nice class of polytopes whose volume has a combinatorial interpretation. Theorem 3.9 (Stanley [26]). Given a poset P we have that (i) the vertices of (P ) are in bijection with characteristic functions of complements of order ideals of P , O (ii) the normalized volume of (P ) is e(P ), where e(P ) is the number of linear extensions of P , O (iii) the Ehrhart polynomial L (P )(m) of (P ) equals the order polynomial Ω(P, m + 1) of P . O O Definition 3.10. Given a poset P and a positive integer m, the order polynomial Ω(P, m) is the number of order preserving maps η : P 1, 2, . . . , m . → { } 3.2. Flow polytopes of planar graphs are order polytopes. The following theorem, which states that a flow polytope of a planar graph is an order polytope, is a result communicated to us by Postnikov [21]. Given a connected graph G with the conventions of Section 3.1, we say that G is planar if it has a planar embedding so that if vertex i is in position (xi, yi) then xi < xj whenever i < j. We denote by G∗ the truncated dual graph of G, which is the dual graph with the vertex corresponding to the infinite face deleted. The orientation of the edges of G induces an orientation 7 G G∗

G 7→ PG G P R

1ˆ S T

1 2 3 4 5 U V W P

P 7→ GP H

0ˆ

Figure 4. Illustration of how to obtain the Hasse diagram PG from a planar graph G (top arrow), and how to obtain a planar graph GP from a strongly planar poset P (bottom arrow).

1ˆ a R b h c S e g T i U V W d f j

0ˆ

Figure 5. Illustration of the maps fl f and f fl from Definitions 3.12 and 3.15 explained in Examples 3.13 and 3.16.7→ 7→

of the edges of G∗ (faces of G) from lower to higher y-coordinates of the end points. This allows us to consider G∗ as the Hasse diagram of a poset that we denote by PG. See Figure 4. Note that by Euler’s formula, #P = #E(G) #V (G) + 1 which equals dim( ). Let P := P 0ˆ, 1ˆ . G − FG G G t { } Theorem 3.11 (Postnikov [21]). Let G be a planar graph on the vertex set [n] such that at each b vertex v [2, n 1] there are both incoming and outgoing edges. Fix a planar embedding of G ∈ − with the above conventions. Then the map ϕ : G (PG) given in Definition 3.12 is an integral int int F → O equivalence. In particular, G (PG) (PG). F ≡ O ≡ O b

Definition 3.12. Define ϕ : G (PG) by fl (f(x))x PG , where f : PG R 0 is given by F →b O 7→ ∈ → ≥ (3.1) b f(x) = fl(e). b e p X∈ The latter sum is taken over the edges e that are intersected by a(ny) path p in PG from 0ˆ to x.

Example 3.13. Given the graph G and the corresponding poset PG in Figure 5,b the map fl f from Definition 3.12 is as follows: 7→ 8 f(R) = fl(b) + fl(c) + fl(d), f(S) = fl(c) + fl(d) f(T ) = fl(i) + fl(j) f(U) = fl(d) f(V ) = fl(f) f(W ) = fl(j). Lemma 4.1 below shows that f in Definition 3.12 is well-defined, while Lemma 4.2 shows that ϕ indeed maps points in to points in (P ). The proof of Theorem 3.11 is given in Section 4. FG O G 3.3. Order polytopes of strongly planar posets are flow polytopes. We now state a converse b of Theorem 3.11, showing that the order polytope of a strongly planar poset is a flow polytope. A poset P is strongly planar if the Hasse diagram of P := P 0ˆ, 1ˆ has a planar embedding with y coordinates respecting the order of the poset. For example,t { the} “bowtie” poset defined by the relations a < c, a < d, b < c, b < d is planar, but not stronglyb planar. Given a strongly planar poset P , let H be the (planar) graph obtained from the Hasse diagram of P with two additional edges from 0 to 1, one of which goes to the left of all the poset elements and another to the right. We can then define the graph GP to be the truncated dual of H. The orientationb of GP is inherited from theb posetb in the following way: if in the construction of the truncated dual, the edge e of GP crosses the edge x y where x < y in P , then y is on the left and x is on the right as you traverse e. See Figure 4. →

Theorem 3.14. If P is a strongly planar poset, then the map φ : (P ) GP given in Definition int int O → F 3.15 is an integral equivalence. In particular, (P ) (P ) G . O ≡ O ≡ F bP Definition 3.15. Define φ : (P ) by (f(x)) fl, where fl : E(G ) is given GP x bPˆ P R 0 by O → F ∈ 7→ → ≥ (3.2) b fl(e) = f(y) f(x), − where e crosses the Hasse diagram edge x y (in the dual construction). → Example 3.16. Given the graph G and the corresponding poset PG in Figure 5, the map f fl from Definition 3.15 is as follows: 7→ fl(a) = 1 f(R) fl(f) = f(V ) − fl(b) = f(R) f(S) fl(g) = f(T ) f(V ) − − fl(c) = f(S) f(U) fl(h) = f(R) f(T ) − − fl(d) = f(U) fl(i) = f(T ) f(W ) − fl(e) = f(S) f(V ) fl(j) = f(W ). −

We postpone the proof of Theorem 3.14 result to Section 4.

4. Proofs of Theorem 3.11 and Theorem 3.14 This section provides the proofs of Theorems 3.11 and 3.14.

Lemma 4.1. Given a flow fl G, the map f : PG R 0 is independent on the path p chosen in (3.1). ∈ F → ≥ 9 b a

0 p1 p 1 v GP v

b P

Figure 6. Left: Local move of paths in the proof of Lemma 4.1. Right: Illustration of why flow is conserved in the map from (P ) to . O FGP

Proof. Let p1 and p2 be two paths in PG from 0ˆ tobx. We show that (4.1) fl(e) = fl(e). b e p1 e p2 X∈ X∈ If p1 and p2 coincide, (4.1) is trivial. We induct on the number of vertices of G enclosed by the two paths p1 and p2. Without loss of generality, assume p1 is left of p2 given the planar drawing of G. Let v be the vertex with the smallest x-coordinate enclosed by the two paths p1 and p2 in the planar drawing of G. By construction, all the incoming edges in G to v are crossed by path p1 and x is not a face between two incoming edges to v. Next, we do the following local move to change the path p1: let p10 be the path that coincides with p1 except that it crosses the outgoing edges of v (see Figure 6). By conservation of flow on vertex v, the sum of the flow of the incoming edges to v equals the sum of the flow of the outgoing edges from v. Since these are the only crossed edges that p1 and p10 differ on we have that fl(e) = fl(e). e p1 e p0 X∈ X∈ 1 The paths p10 and p2 have one fewer vertex of G enclosed by them than the paths p1 and p2. By induction we have fl(e) = fl(e). e p0 e p2 X∈ 1 X∈ Comparing the latter two equations, the result follows. 

Next, we show that given a flow fl in G the point ϕ(fl) = (f(x))x PG is in (PG). F ∈ O Lemma 4.2. Given a flow fl , the image ϕ(fl) (P ). ∈ FG ∈ O G b Proof. Note that f(0)ˆ = 0. We have that 0 f(x) since fl(e) 0 for all edges e. Also, f(x) 1 ≤ b ≥ ≤ since the set of edges whose sum of flows equals f(x) can always be extended to a path from 0ˆ to 1.ˆ By repeated application of Lemma 4.1, the total flow in such a path is 1. Thus, f(1)ˆ = 1. Next, if x0 covers x in PG then there is an edge e0 in G separating the graph faces x and x0. Thus f(x0) = fl(e0) + f(x) f(x). Hence the linear map f takes a point (fl(e))e E(G) of G to the ≥ ∈ F b point (f(x))x PG of the order polytope (PG).  ∈ O Lemma 4.3. Given a point in (P ) viewed as a function f : P , the flow fl : E(G ) b R 0 P R 0 as in Definition 3.15 is in O. → ≥ → ≥ FGP b b Proof. Let f : P R 0 be a point in (P ) and let e be an edge in GP crossing the Hasse diagram → ≥ O edge x y of P . Since x P y then by definition of (P ), fl(e) = f(y) f(x) 0. → b ≤ b 10 O − ≥ b b Next, we find the netflows at each vertex of G. Consider the leftmost (rightmost) path in P from 0ˆ to 1.ˆ This path crosses all the outgoing (incoming) edges in G of vertex 1 (vertex n). We have that b fl(e) = fl(e) = f(1)ˆ f(0)ˆ = 1 0 = 1. − − e E,in(e)=1 e E,fin(e)=n ∈ X ∈ X For an internal vertex v [2, n 1], let a be the face bordering the highest incoming and outgoing edge to v. Similarly, let∈ b be− the face bordering the lowest incoming and outgoing edge to v. Consider the paths pin and pout be the paths in P from b to a crossing the incoming and outgoing edges to v respectively (see Figure 6, Right). Then the total incoming and outgoing flow to vertex v are b fl(e) = (f(w) f(z)) = f(a) f(b), − − e E,fin(e)=v z w in pin ∈ X → X fl(e) = (f(w) f(z)) = f(a) f(b), − − e E,in(e)=v z w in pout ∈ X → X This shows the flow is conserved on vertex v, and thus fl( ) is in . · FGP  Lemma 4.3 shows that φ( (P )) . O ⊂ FGP

Proof of Theorem 3.11 and Theoremb 3.14. Note that given a planar graph G we have that Q := PG is a strongly planar poset and that G = G. Given a flow fl in , let f = ϕ(fl) the corresponding Q FG point in O(PG) and fl0 be the flow φ(ϕ(fl)) = φ(f). Let e be an edge of G crossing the Hasse diagram edge x y in PG b → fl0(e) = f(y) f(x) b − = fl(e ) fl(e ), 1 − 2 e1 p e2 q X∈ X∈ where p is a path PG from 0ˆ to y and q is a path PG from 0ˆ to x. By Lemma 4.1 the value of f(y) is independent of the choice of path, so by letting p = q + x y the last difference becomes fl(e), → showing that fl0(eb) = fl(e). A similar argument showsb that ϕ φ is the identity. Thus the maps φ and ϕ are inverses of each other and they both preserve integer◦ points. Therefore, φ and ϕ are int integral equivalences. Using Lemma 3.8 giving (P ) (P ) for any poset P we are done. O ≡ O  Remark 4.4. By Theorem 3.11, if G is a planarb graph then G is integrally equivalent to an order polytope. This raises the question of whether this relation holdsF for non-planar graphs: for instance for the polytope (n) = for n 4. We can use a similar construction to that in the CRY ∼ FKn+1 ≥ proof of Theorem 3.11 to show that K5 and K6 are integrally equivalent to the order polytopes of the posets: F F

, We leave it as a question whether (dimension 15, 32 vertices, volume 140) is an order polytope. FK7 11 Figure 7. Examples of Young diagrams, their associated planar posets P and graphs G such that the order polytope (P ) and are integrally equivalent. P O FGP 5. (n) and the family of polytopes (ASM) ASMCRY F In this section, we introduce the ASM-CRY family of polytopes (ASM), which includes (n), and show that each of these polytopes is a face of theF ASM polytope. We, fur- thermore,ASMCRY show that each polytope in this family is both an order and a flow polytope. Then, using the theory of order polytopes as discussed in Section 3.1, we determine their volumes and Ehrhart polynomials.

Definition 5.1. Let δn = (n 1, n 2,..., 2, 1) be the staircase partition considered as the positions (i, j) of an n n matrix given− by− (i, j) j i 1 . Let the partition λ = (λ , λ , . . . , λ ) δ × { | − ≥ } 1 2 k ⊆ n denote matrix positions (i, j) 1 i k, n λi + 1 j n, λi n i . We define the ASM-CRY{ family| ≤ ≤ − ≤ ≤ ≤ − } (ASM)(n) := (n) λ δ , F {Pλ | ⊆ n} where (n) := (a )n (n) a = 0 for i j 2 and for (i, j) λ . Pλ ij i,j=1 ∈ A | ij − ≥ ∈ Note that (n) =  (n), as in Definition 2.5. P∅ ASMCRY In the following proposition we give a convex hull description of the polytopes in this family.

Proposition 5.2. The polytope λ(n) (ASM)(n) is the convex hull of the n n alternating sign matrices (A )n with A =P 0 for∈i F j 2 and for (i, j) λ. × ij i,j=1 ij − ≥ ∈ Proof. Let (n) denote the convex hull of the n n alternating sign matrices (A )n with Qλ × ij i,j=1 Aij = 0 for i j 2 and for (i, j) λ. It is easy to see that λ(n) is contained in λ(n), since matrices in both− polytopes≥ have the∈ same prescribed zeros and satisfyQ the inequality descriptionP of the full ASM polytope (n). A It remains to prove that λ(n) is contained in λ(n). Suppose there exists a matrix b = n P Q (bij)i,j=1 λ(n) such that b / λ(n). We know that b is in the convex hull of all n n ASMs. ∈1 P 2 ∈k Q 1 k × So b = µ1A + µ2A + + µkA , where A ,...A are distinct n n alternating sign matrices and ··· 1 × 1 µ1, . . . , µk > 0. At least one of these ASMs, say A must have a nonzero entry Aij for some (i, j) satisfying either i j 2 or (i, j) λ. Suppose i j 2; the argument follows similarly in the − ≥ ∈ 1 − ≥ 2 2 case (i, j) λ. Now since bij = 0 and Aij = 0, there must be another ASM, say A such that Aij ∈ 1 6 2 is nonzero of opposite sign. Say Aij = 1 and Aij = 1. Then by the definition of an alternating 2 − 3 sign matrix, there must be j0 < j such that Aij0 = 1. But bij0 = 0 as well, so there must be an A 3 3 with A 0 = 1 and j < j such that A 00 = 1. Eventually, we will reach the border of the matrix ij − 00 0 ij and reach a contradiction. Thus, (n) = (n). Pλ Qλ  We show in Theorem 5.3 below that the polytopes in (ASM)(n) are faces of (n). First, we need some terminology from [29]. Consider n2 + 4n verticesF on a square grid: n2 ‘internal’A vertices (i, j) and 4n ‘boundary’ vertices (i, 0), (0, j), (i, n + 1), and (n + 1, j), where 1 i, j n. Fix the 12 ≤ ≤ orientation of this grid so that the first coordinate increases from top to bottom and the second coordinate increases from left to right, as in a matrix. The complete flow grid Cn is defined as the directed graph on these vertices with directed edges pointing in both directions between neighboring internal vertices within the grid, and also directed edges from internal vertices to neighboring border vertices. That is, C has edge set ((i, j), (i, j 1)), ((i, j), (i 1, j)) i, j = 1, . . . , n .A simple n { ± ± | } flow grid of order n is a subgraph of Cn consisting of all the vertices of Cn, and in which four edges are incident to each internal vertex: either four edges directed inward, four edges directed outward, or two horizontal edges pointing in the same direction and two vertical edges pointing in the same direction. An elementary flow grid is a subgraph of Cn whose edge set is the union of the edge sets of some simple flow grids. See Figure 8. Theorem 5.3. The polytope (n) (ASM)(n) is a face of (n), of dimension n λ . In Pλ ∈ F A 2 − | | particular, (n) = (n) is a face of (n), of dimension n . P∅ ASMCRY A 2
Proof. In Proposition 4.2 of [29], it was shown that the simple flow grids
of order n are in bijection with the n n alternating sign matrices. In this bijection, the internal vertices of the simple flow grid correspond× to the ASM entries; the sources correspond to the ones of the ASM, the sinks correspond to the negative ones, and all other vertex configurations correspond to zeros. In Theorem 4.3 of [29], it was shown that the faces of (n) are in bijection with n n elementary flow A × grids, with the complete flow grid Cn in bijection with the full ASM polytope (n). This bijection was given by noting that the convex hull of the ASMs in bijection with all theA simple flow grids contained in an elementary flow grid is, in fact, an intersection of facets of the ASM polytope (n), A and is thus a face of (n). Since, by Proposition 5.2, λ(n) equals the convex hull of the ASMs in it, we need only showA there exists an elementary flowP grid whose contained simple flow grids correspond exactly to these ASMs. We can give this elementary flow grid explicitly. We claim that the directed edge set S := (i,j) Si,j where S ((i, j), (i, j 1)) , ((i, j), (i + 1, j)) if i j 2 { − } − ≥ Si,j := ((i, j), (i, j + 1)) , ((i, j), (i 1, j)) if (i, j) λ { − } ∈  ((i, j), (i, j 1)) , ((i, j), (i 1, j)) otherwise { ± ± } is the union of the directed edge sets of all the simple flow grids in bijection with ASMs in λ(n).  P It is clear that the directed edge set of any simple flow grid corresponding to an ASM in λ(n) is in S; it remains to show that any edge in S appears in some simple flow grid. Note thatP the directed edges listed in the first two cases appear in every simple flow grid in bijection with an ASM in λ(n). For the remaining edges, note that if Aij = 1, then in the corresponding simple flow grid,PS = ((i, j), (i, j 1)) , ((i, j), (i 1, j)) . It is easy to construct a permutation matrix i,j { ± ± } A λ(n) with Aij = 1 for any fixed (i, j) with i j < 2 and (i, j) / λ. Thus the digraph with the edge∈ P set S is an elementary flow grid. Furthermore,− no other simple∈ flow grid can be constructed from directed edges in this set, since such a simple flow grid would have to include an edge pointing in the wrong direction in either the region i j 2 or (i, j) λ. Thus, (n) is a face of (n). − ≥ ∈ Pλ A To calculate the dimension of λ(n), we use the following notion from [29]. A doubly directed region of an elementary flow gridP is a connected collection of cells in the grid completely bounded by double directed edges but containing no double directed edges in the interior. Theorem 4.5 of [29] states that the dimension of a face of (n) equals the number of doubly directed regions in the corresponding elementary flow grid. TheA number of doubly directed regions in the elementary 2 n 1 n flow grid corresponding to (n) equals (n 1) − + λ = λ . See Figure 8. Pλ − − 2 | | 2 − | |  Our main result regarding (ASM)(n) is Theorem 1.1,
which
we
prove below. It requires the F following definition (see Figure 9 for examples); also, recall from Section 3.3 the definition of GP . 13 (a) (b) (c)

Figure 8. (a) The complete flow grid C5, which corresponds to the full ASM poly- tope (5). (b) The elementary flow grid corresponding to (5) with λ = (2, 1, 1). A Pλ Note there are six doubly directed regions, thus λ(5) is a face of (5) of dimension six. (c) A simple flow grid which corresponds toP a 5 5 ASM andA is contained in the elementary flow grid of (b). ×

Definition 5.4. Let δ and λ δ be as in Definition 5.1. Let (δ λ) be the poset with elements n ⊆ n n \ ∗ p corresponding to the positions (i, j) δ λ with partial order p p 0 0 if i i and j j . ij ∈ n \ ij ≤ i j ≥ 0 ≤ 0 We now prove Theorem 1.1 by first establishing two lemmas to show that (n) is integrally Pλ equivalent to the order polytope of the poset (δn λ)∗. Then since this poset is strongly planar, by Theorem 3.14 its order polytope is integrally equivalent\ to the flow polytope . FG(δn\λ)∗ Given a matrix (a )n (n), define the corner sum matrix (c )n by ij i,j=1 ∈ Pλ ij i,j=1

cij = ai0j0 . 1 i0 i, j ≤Xj0≤n ≤ ≤ For S R, let (δn λ, S) be the set of functions g : δn λ S. We view the order polytope ⊆ A \ \ → of (δn λ)∗ as a subset of (δn λ, [0, 1]). Define Ψ : λ(n) (δn λ, R) by a ga where g (i, j)\ = 1 c . See the secondA \ map in Figure 10. P → A \ 7→ a − ij Lemma 5.5. The image of Ψ is in (δ λ, [0, 1]), i.e. if a g then g (i, j) = 1 c [0, 1]. A n \ 7→ a a − ij ∈ Proof. We first show that cij 0 for all i and j. By the defining inequalities of the ASM polytope (n) (see Definition 2.4), we≥ have that the partial row and column sums of any a (n) satisfy theA following for each fixed 1 i, j n: ∈ A ≤ ≤ i n (5.1) ai0j 0 and aij0 0. 0 ≥ 0 ≥ iX=1 jX=j i n Since c = 0 0 a 0 0 and the interior sum is nonnegative by (5.1), c 0 as desired. ij i =1 j =j i j ij ≥ Next we show that for a (n), c 1 for all j > i 1. (Note this is not true for all matrices P P ∈ Pλ ij ≤ ≥ in (n); for example the permutation matrix corresponding to 4321 has c23 = 2.) FirstA note c 1 for all j, since by (5.1) each a 0 and n a = 1. 1j ≤ 1j ≥ j=1 ij Now fix j > i 2. We have ≥ P n i n i j 1 i j 1 i − − cij = ai0j0 = ai0j0 ai0j0 = i ai0j0 0 0 0 0 − 0 0 − 0 0 jX=j iX=1 jX=1 iX=1 jX=1 iX=1 jX=1 iX=1 14 since the sum of each row is 1. Note j 1 i i 1 i j 1 i − − − ai0j0 = ai0j0 + ai0j0 . 0 0 0 0 0 0 jX=1 iX=1 jX=1 iX=1 jX=i iX=1 Now i 1 i i 1 n i 1 − − − ai0j0 = ai0j0 = 1 = i 1 0 0 0 0 0 − jX=1 iX=1 jX=1 iX=1 jX=1 since ai0j0 = 0 for j0 < i < i0. So j 1 i j 1 i − − cij = i (i 1) ai0j0 = 1 ai0j0 . − − − 0 0 − 0 0 jX=i iX=1 jX=i iX=1 i j 1 i But by (5.1), 0 a 0 0 0, so 0− 0 a 0 0 and thus c 1 for all j > i. i =1 i j ≥ j =i i =1 i j ij ≤ We therefore have that 0 c 1 for all j > i, so that 0 g (i, j) 1 as desired. P ≤ ijP≤ P ≤ a ≤  Lemma 5.6. The image of Ψ is in the order polytope ((δ λ) ). O n \ ∗ Proof. By Lemma 5.5 we know that the image of Ψ is in (δ λ, [0, 1]). Note that if i i and A n \ 0 ≤ j0 j, then cij ci0j0 , thus we have that ga(i, j) ga(i0, j0) if and only if (i, j) (i0, j0) in (δn λ)∗. So≥g is in the order≥ polytope ((δ λ) ). ≤ ≤ \ a O n \ ∗  Proof of Theorem 1.1. By Lemmas 5.5 and 5.6 we have that the map Ψ is an affine transformation from (n) to ((δ λ) ) of the form a 1 Aa where A is a 0, 1-upper unitriangular matrix. Pλ O n \ ∗ 7→ − Thus, Ψ is a bijection between λ(n) and ((δn λ)∗) that preserves their respective lattices. This shows that the two polytopes areP integrallyO equivalent.\ Finally since the poset (δn λ)∗ is strongly planar, by Theorem 3.14 λ(n) is also integrally equivalent to the flow polytope\ . P FG(δn\λ)∗  By Stanley’s theory of order polytopes [26] (see Theorem 3.9) we express the volume and Ehrhart polynomial of the polytopes in this family in terms of their associated posets. Recall that e(P ) denotes the number of linear extensions of the poset P . Corollary 5.7 ([26]). For (n) in (ASM)(n) we have that its normalized volume is Pλ F vol( (n)) = e ((δ λ)∗) , Pλ n\ and its Ehrhart polynomial is

∗ L λ(n)(t) = Ω(δn λ) (t + 1). P \ Note that using Theorem 3.4 and the discussion below it, we can express the volume and Ehrhart polynomial of any flow polytope as a Kostant partition function. Thus, Theorem 1.1 gives us several Kostant partition function identities. Corollaries 1.2, 5.10 and 5.11 compute the volumes and Ehrhart polynomials of three subfamilies of polytopes in (ASM)(n) that are associated to posets with a nice number of linear extensions and vertices. ThisF includes the ASM-CRY polytope. See Figure 9.

Proof of Corollary 1.2. When λ = ∅, (n) is integrally equivalent to the order polytope δ∗ of P∅ O n the poset δn∗ (that is, the type An 1 positive root lattice). By Theorem 3.9 the number of− vertices and volume of (n) are given by the number of order P∅ ideals and linear extensions of the poset δn∗ respectively. Next we compute each of these. The order ideal of the poset δn∗ correspond to shapes λ δn which in turn correspond to Dyck 1 2n ⊆ paths counted by the Catalan number Cn = n+1 n . 15
Polytope Shape Poset # Vertices Volume

λ # order ideals #SYT (δn/λ) λ(n) of the poset = # linear extensions (δ /λ)∗ P n of the poset (δn/λ)∗

(n) n ASMCRY 1 2n (2)! (i.e. ∅(n)) n+1 n 1n 13n 2 (2n 3)1 P − − ··· −

δn 2(n) F2n 1 E2n 3 P − − −

n 1 δn 1(n) 2 − (n 1)! P − −

Figure 9. Some polytopes in the family (ASM)(n) and their corresponding num- bers of vertices and volumes; see TheoremF 1.1 and Corollaries 1.2, 5.7, 5.10, and 5.11. ‘Shape’ refers to the entries in the matrix not fixed to be zero. All diagrams are drawn in the case n = 5.

.3 .4 .1 .2 .7 .3 .2 .8 .7 -.2 -.1 .4 ci,j .6 .6 1 ci,j − .7 .4 0 .8 .1 .1 .7 ga 0 0 .9 .1 .3 .4 .3

Figure 10. A map from a point in (4) to a point in the order polytope. First, take the northeast corner sumASMCRY of each entry above the main diagonal. Then subtract that value from 1.

The number of linear extension of this poset is the number of standard Young tableaux (SYT) of shape δ = (n 1, n 2,..., 2, 1). Thus n − − n ! vol (n) = #SYT (δ ) = 2 , P∅ n 1n 13n 2 (2n 3)1 − − ···
− where the second equality follows by using the hook-length formula [27, Cor. 7.21.6] to compute this number of tableaux. Lastly, by Theorem 3.9 L (t) = Ω ∗ (t + 1). When t is an integer, Ω ∗ (t + 1) counts the the ∅(n) δn δn number of plane partitions ofP shape δ with largest part t. By a result of Proctor [22] (see also n ≤ [12]) this number is given by the product formula in the RHS of (1.1). 

Since the polytope (n) is contained in (n) then we can bound the volume and number of lattice pointsCRY of the former with theASMCRY corresponding volume and number of lattice points of the latter.

16 Corollary 5.8. For n 1 and t N we have that ≥ ∈ n 2 − Cat(i) #SYT (δ ) ≤ n Yi=1 2t + i + j 1 L (n)(t) − . CRY ≤ i + j 1 1 i

Remark 5.12. For the case λ = δn k, the polytope δn−k (n) is integrally equivalent to the order − P polytope of the poset (δn δn k)∗. The number of vertices of the polytope (order ideals of the poset) is given by the number\ − of Dyck paths with height at most k [24, A211216], [14, 3.1]. The § volume of the polytope is given by the number of skew SYT of shape δn/δn k. There are formulas for this number of SYT as determinants of Euler numbers (e.g see Baryshnikov-Romik− [4]). We now turn from our investigation of the family of polytopes (ASM)(n) to triangulations of flow and order polytopes. F

6. Triangulations of flow polytopes of planar graphs As we have seen in Section 3, flow polytopes of planar graphs are integrally equivalent to order polytopes. In this section we relate a known triangulation of flow polytopes by Danilov–Karzanov– Koshevoy and a well known triangulation of order polytopes. 17 6.1. Canonical triangulation of order polytopes. Recall that vertices of an order polytope (P ) correspond to characteristic functions of order filters (i.e. complements of order ideals). OStanley [26] gave a canonical way of triangulating the order polytope (P ) for an arbitrary poset O P . Namely, for a linear extension (a1, a2, . . . , am) of the poset P on elements a1, a2, . . . , am , define the simplex { } (6.1) ∆ := (x , . . . , x ) [0, 1]m x x x . a1,a2,...,am { 1 m ∈ | a1 ≤ a2 ≤ · · · ≤ am } Note that the m + 1 vertices of this simplex are 0, 1 vectors whose 0-coordinates are indexed by length k prefixes a1, . . . , ak of the linear extension for k = 0, 1, . . . , m. The simplices ∆a1,a2,...,am corresponding to all linear extensions of P are top dimensional simplices in a triangulation of (P ), which we refer to as the canonical triangulation of (P ). There are also two established combinatorialO ways of triangulating flow polytopes: one givenO by Postnikov and Stanley (PS) [21, 25] (defined in Section 7.1), and one by Danilov, Karzanov and Koshevoy (DKK) [9] (defined in Section 6.3). All the aforementioned triangulations are unimodular. The goal of this section is to relate the DKK triangulation of flow polytopes of planar graphs and Stanley’s linear extension triangulation of the corresponding order polytope. As before, it will be more convenient for us to work with the integrally equivalent polytope int (P ) (P ) and the integral equivalence ν from Lemma 3.8. The canonical triangulation of O ≡ O 1 1 (P ) maps under ν− to the canonical triangulation of (P ). We will denote ν− (∆a1,a2,...,am ) by Ob O ∆a1,a2,...,am . Of course: b (6.2)b ∆ := (x , x , . . . , x , x ) [0, 1]m+2 0 = x x x x x = 1 . a1,a2,...,am { 0ˆ 1 n 1ˆ ∈ | 0ˆ ≤ a1 ≤ a2 ≤ · · · ≤ am ≤ 1ˆ } In this section, we show that given a planar graph G, the canonical triangulation of (P ) maps to b G a DKK triangulation of via the integral equivalence φ from Theorem 3.14. ThisO result was first FG observed by Postnikov [21]. We also construct a direct bijection between linear extensionsb of PG, which index the canonical triangulation of (PG), and maximal cliques of G, which index the DKK triangulation of . In Section 7, we proveO for a general graph G that the DKK triangulations of FG are framed Postnikov-Stanley triangulationsb of . In particular, the canonical triangulation FG FG of (PG) for a planar graph G maps to a framed Postnikov-Stanley triangulation of G under integralO equivalence φ from Theorem 3.14. F Inb the following subsection we review the results of Danilov, Karzanov and Koshevoy [9]. 6.2. Danilov–Karzanov–Koshevoy triangulation of flow polytopes. Let G be a connected graph on the vertex set [n] with edges oriented from smaller to bigger vertices. Recall from Propo- sition 3.3, that vertices of G are given by unit flows along maximal directed paths from the source 1 to the sink n. FollowingF [9], we call such maximal paths routes. The following definitions also follow [9]. Let v be an inner vertex of G whenever v is neither a source nor a sink. Fix a framing at each inner vertex v, that is, a linear ordering on the ≺in(v) set of incoming edges in(v) to v and the linear ordering on the set of outgoing edges out(v) ≺out(v) from v.A framed graph, denoted by (G, ), is a graph G with a framing at each inner vertex. For a framed graph G and an inner vertex v≺, we denote by In(v) and by Out≺(v) the set of maximal paths ending in v and the set of maximal paths starting at v, respectively. We define the order on the paths in In(v) as follows. If P,Q In(v), P = Q, then let w be the unique vertex ≺In(v) ∈ 6 after which P and Q coincide and before which they differ. Let eP be the edge of P entering w and e be the edge of Q entering w. Then P Q if and only if e e . Similarly, if Q ≺In(v) P ≺in(w) Q P,Q Out(v), P = Q, then let w be the unique vertex before which P and Q coincide and after ∈ 6 which they differ. Let eP be the edge of P leaving w and eQ be the edge of Q leaving w. Then P Out(v) Q if and only if eP out(w) eQ. ≺ ≺ 18 Given a route P with an inner vertex v, denote by P v the maximal subpath of P ending at v and by vP the maximal subpath of P starting at v. We say that the routes P and Q are coherent at a vertex v which is an inner vertex of both P and Q if the paths P v, Qv are ordered the same way as vP, vQ; that is, P v In(v) Qv if and only if vP Out(v) vQ. We say that routes P and Q are coherent if they are coherent≺ at each common inner vertex.≺ We call a set C of mutually coherent routes a clique. Let max(G, ) be the set of maximal cliques (with respect to number of routes) of the framed graph GC. ≺

Definition 6.1. Given a framed graph G, and a clique C of the framed graph G, denote by ∆C the convex hull of the vertices of corresponding to the unit flows along routes in the clique C. FG Theorem 6.2 below is a special case of [9, Theorems 1 & 2]. Theorem 6.2. [9, Theorems 1 & 2] Given a framed graph (G, ), the set of simplices ≺ ∆ C max(G, ) , { C | ∈ C ≺ } corresponding to maximal cliques of the framed graph G are the top dimensional simplices in a regular unimodular triangulation of G. Moreover, lower dimensional simplices ∆C of this trian- gulation are obtained as convex hullsF of the vertices corresponding to the routes in non-maximal cliques of G. C We call the triangulations specified in Theorem 6.2 the Danilov-Karzanov-Koshevoy (DKK) triangulations of . Each such triangulation comes from a particular framing of the graph. We FG are now ready to prove that the canonical triangulation of (P ) is integrally equivalent to a DKK O G triangulation of G via the map φ : (PG) G from Theorem 3.14. We now define the framing needed for this result.F Consider a planarO graph→ F G on theb vertex set [n] with a particular planar embedding so that if vertex i is in positionb (xi, yi) then xi < xj whenever i < j. At each vertex v [2, n 1] of G there is a natural order on the edges coming from the planar drawing of the graph:∈ order− the incoming edges as well as the outgoing edges top to bottom in increasing order; by top to bottom we mean that if we put a small enough circle C centered at vertex i so that all incoming and outgoing edges to vertex i intersect the circle, then we order the incoming (and outgoing) edges top to bottom by decreasing y coordinates of their intersection with the circle C. We call this framing the planar framing of G, to emphasize that this framing comes from a particular planar embedding of the graph G.

6.3. The canonical triangulation of (PG) is integrally equivalent to a DKK triangula- tion of . We are now ready to stateO the main result of this section. FG b Theorem 1.3. Given a planar graph G, the canonical triangulation of (PG) maps to the Danilov- Karzanov-Koshevoy triangulation of coming from the planar framingO via the integral equivalence FG map φ : (P ) given in Theorem 3.14. b O G → FG We proveb Theorem 1.3 together with Theorem 6.6 below.

Recall that by Theorem 3.9 vertices of (PG) are in bijection with order ideals of PG – indeed the vertices of (P ) are the characteristicO functions of the complements of the order ideals in the O G poset P . By Lemma 3.8 the vertices of (P ) are also naturally indexed by the order ideals of G O G PG. Let fI be the vertex of (PG) indexed by the order ideal I of PG. Given a planar graph G we say that a route R of G separatesO the orderb ideal I and the complement P I if the elements G \ of PG below the route R in theb planar drawing of G and the truncated dual PG are exactly the elements of the order ideal I. 19 Proposition 6.3. Given a vertex fI of (PG) indexed by the order ideal I of PG we have that φ(f ) is the unit flow along the route R inOG separating I and P I. Moreover, any route R in G I G \ separates some order ideal I and P I. b G \ Proof. As explained in Section 3, the elements of PG correspond to bounded regions defined by G. Given an indicator function fI for the complement of an order ideal I of this poset, by Definition 3.15 the flow φ(fI ) is the specified unit flow. See Figure 11 for an example. 

Next, we define the map Φ∆ between linear extensions of PG, which index the top dimensional simplices in the canonical triangulations of O(PG), and sets of routes corresponding to the vertices of top dimensional simplices in a DKK triangulation of (the latter is shown in Theorem 6.6). FG b Definition 6.4. Given a linear extension a = a1 am of PG, let Φ∆(a) be the following set of routes of G determined by the order ideals whose elements··· are the letters in the prefixes of a,

Φ∆(a) := φ(f a1, ,ak ) k = 0, 1, . . . , m . { { ··· } | } That is, Φ∆(a) is the set of routes of G separating each of the order ideals formed from letters in the prefixes of the linear extension a1, . . . , am. See Figure 11 for an example.

CABD

G CAB D CA PG B C A C ∅

Figure 11. On the left is the planar graph G and the poset PG on elements A, B, C, D. On the right are all prefixes of the linear extension CABD of PG – each of which corresponds to an order ideal of PG –, which specify the 0 coordinates of the vertices of (PG), and the routes these vertices correspond to under the map φ. Note that the fiveO resulting routes form a maximal clique in G with respect to the planar framing thatb orders both the incoming and outgoing edges top to bottom in increasing order. Under Φ∆ the linear extension CABD is mapped to the maximal clique in G formed by the five routes on the right.

Next, we show that the routes in Φ∆(a) form a clique. Lemma 6.5. For a planar graph G, fix a linear extension a = a a of P indexing a simplex 1 ··· m G ∆a1 am of O(PG). Then the routes in Φ∆(a) are pairwise coherent in the planar framing of G. ···

Proof. Let v1 and v2 be vertices of ∆a1 am mapping to routes P1 and P2 under φv. It suffices to b b ··· show that P1 and P2 are coherent in the planar framing of G. Let the coordinates of v equal to 0 be x , x , . . . , x and the coordinates of v equal to 0 1 b 0ˆ a1 ak1 2 be x , x , . . . , x and assume without loss of generality that k < k . Since both a , . . . , a 0ˆ a1 ak2 1 2 1 k1 and a1, . . . , ak2 are prefixes of the linear extension of a1, . . . , am, we see that the upper boundary of the regions corresponding to a1, . . . , ak1 lies weakly below that of the boundary of the regions corresponding to a1, . . . , ak2 , and thereby the corresponding routes P1 and P2 are coherent with respect to the planar framing.  20 Theorem 6.6. Given a planar graph G, the map Φ∆ defined above is a bijection between linear extensions of PG and maximal cliques in G in the planar framing. Proof of Theorems 1.3 & 6.6. By Theorem 3.14, φ is an integral equivalence between (P ) and O G G. In particular, the polytopes (PG) and G are of the same dimension and same relative volume.F Therefore, the top dimensionalO simplicesF in their respective triangulations haveb the same number of vertices, and the numberb of simplices in any of their unimodular triangulations are the same. Thus, to show Theorems 1.3 & 6.6 it suffices to show that φ restricts to a bijection on the vertices of (PG) and G and that the set of routes that Φ∆ maps a linear extension to are pairwise coherent inO the planarF framing. The former is follows from Proposition 6.3, while the latter from Lemma 6.5.b 

Corollary 6.7. Given a planar graph G, the number of linear extensions of PG equals the number of maximal cliques in G in any framing. Proof. The statement is immediate from Theorem 1.3 for the planar framing. But since the Danilov- Karzanov-Koshevoy triangulations are unimodular, the number of maximal cliques in G is inde- pendent of the framing. 

7. Triangulations of flow polytopes of general graphs

In Theorem 6.6 we gave a bijection from linear extensions of PG to maximal cliques of G in the planar framing. In this section we will see that given any two framings of a graph G (not necessarily planar) there is a natural bijection between their sets of maximal cliques. Therefore, combining the bijection from Theorem 6.6 and the one just mentioned, we obtain a bijection between linear extensions of PG and maximal cliques in any framing of a planar graph G. More generally, this section is devoted to studying the set of DKK triangulations of a flow polytope and the framed Postnikov-Stanley (PS) triangulations of , which we define in this FG FG section. We show that the set of DKK triangulations of a flow polytope G is equal the set of framed PS triangulations of . As a consequence of our proof, we obtain aF bijection between the FG objects indexing the PS triangulation of a flow polytope G, namely, nonnegative integer flows on n 1 F the graph G with netflow (0, d2, . . . , dn 1, i=2− di), where di is the indegree of vertex i in G minus 1 − − 1 , and the objects indexing the DKK triangulation of a flow polytope G, namely, maximal cliques in a fixed framing of G. This answers Postnikov’sP question [21] aboutF a bijection between the sets indexing the maximal simplices of both triangulations. We also obtain a natural bijection between the sets of maximal cliques of G in different framings, as mentioned in the previous paragraph.

7.1. Framed Postnikov-Stanley triangulations. We now define framed Postnikov-Stanley tri- angulations. These triangulations were used in [16], though they were not described explicitly there, and we follow closely the exposition therein. A bipartite noncrossing tree is a tree with left vertices x1, . . . , x` and right vertices x`+1, . . . , x`+r with no pair of edges (xp, x`+q), (xt, x`+u) where p < t and q > u. We denote by , the set of TI O bipartite noncrossing trees where and are the ordered sets (x1, . . . , x`) and (x`+1, . . . , x`+r) I `+r O2 respectively. We have that # , = ` −1 , since the elements of , are in bijection with weak TI O − TI O compositions of ` 1 into r parts. A tree T in , corresponds to the composition (b1, . . . , br) of
I O (indegrees 1), where− b denotes the number of edgesT incident to the right vertex x in T minus 1. − i `+i Example 7.1. The bipartite tree in Figure 13 corresponds to the composition (1, 0, 2).

1See Definition 3.1 and the discussion in Section 3.1 for the relation of nonnegative integer flows with a given netflow vector to Kostant partition functions as well as Theorem 3.4. 21 f G f1 G e 1 e1 1 1 1 1 1 1 f 1 2 e2 2 2 3 e2 f3 2 3 3 e3 0 e3 2 4 2 4 3 e4 f3 e4 f2 2 0 0 0 e e1 f1 1 f1 e2 e2 f (2) f2 (2) 2 e3 G e1 + f1 e3 GT f3 T f3 e1 + f1 e4 e2 + f1 e4 e2 + f3 e2 + f1 (1,0,2) T (1,0,2) e3 + f3 T e2 + f2 e4 + f3 e2 + f3 e2 + f2 e3 + f3

e4 + f3 (a) (b)

Figure 12. Replacing the incident edges of vertex 2 in a graph H, by a noncrossing tree T encoded by the composition (1, 0, 2) of 3 = indegH (2) 1 using two different framings (indicated by the blue numbers incident to vertex 2− in G): (a) the framing is increasing top to bottom, (b) different framing.

e1 + f1 e1 f1 + f1 e e2 e 2 2 + f2 e f2 e 2 3 + f e3 + 3 e4 f3 f3 e4 + f3

Figure 13. Based on the noncrossing tree: S(f1) = e1 +f1, e2 +f1 ,S(f2) = e2 + f , and S(f ) = e + f , e + f , e + f . The local{ orderings of these} edges at{ the 2} 3 { 2 3 3 3 4 3} vertices to which they are incoming are e1+f1 < e2+f1 and e2+f3 < e3+f3 < e4+f3.

We now define what we mean by a reduction at vertex i of a framed graph G on the vertex set [n]. Let denote the multiset of incoming edges and the multiset of outgoing edges of i. In Ii Oi addition, we assume that i and i are linearly ordered according to the framing of G. A reduction performed at i of G resultsI in severalO new graphs indexed by bipartite noncrossing trees on the left vertex set I and right vertex set . We define these new graphs precisely below. i Oi Consider a tree T i, i . For each tree-edge (e1, e2) of T where e1 = (r, i) i and e2 = (i, s) , let e + e ∈be T theI O following edge: ∈ I ∈ Oi 1 2 (7.1) e1 + e2 = (r, s).

We call the edge e1 + e2 the sum of edges. Alternatively you can consider this as a path in G consisting of edges e1 and e2. Inductively, we can also define the sum of more than two consecutive edges. (i) Given T in i, i , let G be the graph obtained from G by removing the vertex i and all the TI O T edges of G incident to i and adding the multiset of edges e1 + e2 (e1, e2) E(T ) . See Figures (i) {{ | ∈ }} 12, 15 and 14 for examples of GT . Given a tree T in i, i , a reduction of G at the vertex i with respect to T replaces G (i)TI O by the graphs in GT defined above. The reduction also keeps track which sum of the edges of G is each edge of the new graphs (allowing for the sum of only one element, when an edge was left intact). 22 (i) We now define an inheritance framing of GT for T in i, i , which it inherits from the framing of G as follows: TI O (i) The edges incident to a vertex j smaller than i in G(i) are in bijection with edges incident to vertex j in G. We order the edges in G(i) in the same way as they are ordered in G. (ii) For each vertex j greater than i the multiset of outgoing edges (G(i)) equals (G). We Oj T Oj order these the edges of (G(i)) the same way the edges (G) are ordered. Oj T Oj (iii) For each vertex j greater than i, if j(G) = m1, . . . , mk (the multiset linearly ordered I { (i) } according to the framing of G), then the multiset j(GT ) consists of edges that are sums of edges of G (potentially the empty sum) with anI edge of (G). Thus denote by S(m ), Ij l l [k], the edges in (G(i)) which are sums of edges of G (potentially the empty sum) ∈ Ij T with ml. Then let any edge in S(mp) be less than any edge in S(mq) for p < q, p, q [k]. We now specify the ordering of the edges within the sets S(m ), l [k]. If S(m ) = ∈m l ∈ l { l} then there is nothing to specify. If S(ml) = ml , then draw T with the left and right sets of vertices ordered vertically following the6 linear{ } order of and from the framing of G. Ii Oi We order the edges in S(ml) following the order on the edges of the noncrossing bipartite tree T when viewed from top to bottom (smallest edge to largest). See Figure 13 for an example. Next we describe what we refer to as the framed Postnikov-Stanley (PS) triangulations of G. GivenF a framed graph (G, ) on the vertex set [n], and a nonnegative integer flow ifl( ) on ≺ · G with netflow (0, d2, d3, . . . , dn 1, i di), where di = indegi(G) 1, we explain how to ob- (G, ) − − − tain a simplex ∆ ≺ , such that as ifl runs over all nonnegative integer flow G with netflow ifl P (G, ) (0, d2, d3, . . . , dn 1, i di) we obtain a set of simplices ∆ ≺ that are the top dimensional sim- − − ifl plices of a triangulation of G. It is this triangulation that we term the framed Postnikov-Stanley (PS) triangulation. P F Given a framed graph (G, ) on the vertex set [n], and a nonnegative integer flow ifl( ) on G ≺ · with netflow (0, d2, d3, . . . , dn 1, i di) we read off the nonnegative integer flow values specified − − by ifl( ) on the edges of 2(G) yielding a composition (c1, . . . , c# 2(G)) of d2 = # 2(G) 1. · O O I − Here c corresponds to the flow valueP on the jth largest edge in (G) in the framing. Using j O2 this composition (c1, . . . , c# 2(G)) of d2 = # 2(G) 1 , we build a bipartite tree T2 in 2, 2 as follows. The sets and Ohave an orderingI in the− framing of G. Assume that this orderingTI O is I2 O2 2 = v1 < < v# 2 and 2 = w1 < < w# 2 . Draw the bipartite tree T2 in 2, 2 with I { ··· I } O { ··· O } TI O the left vertex set 2 = v1 < < v# 2 so that the vertices v1 < < v# 2 are ordered top I { ··· I } ··· I to bottom vertically on the left. Similarly, the right vertices w1 < < w# 2 are drawn top to ··· O bottom vertically on the right. We let the degree of vertex wj in T2 be cj + 1. The above uniquely determines the noncrossing bipartite tree T on left and right vertex sets and . See Figure 12 2 I2 O2 for an example. With tree T constructed, we do a reduction at vertex 2 to obtain G := G(2) with 2 2 T2 an inheritance framing. Recursively, given Gi 1, we read off the integer flow values from ifl( ) on the edges of i(G) = − · O i(Gi 1). These flow values can be seen as components of a composition (c1, . . . , c# i(G)) of O − O di + ifl(e) = # i(Gi 1) 1. I − − e,fin(Xe)=i The component cj corresponds to the flow value on the jth largest edge in i(G) = i(Gi 1) in the O O − framing. From this composition (c1, . . . , c# i(G)) we build a bipartite tree Ti in i(Gi− ), i(Gi− ) O TI 1 O 1 as follows. The sets i(Gi 1) and i(Gi 1) have an ordering in the inheritance framing of Gi 1. I − O − − Assume that this ordering is i(Gi 1) = v1 < < v# i(Gi− ) and i(Gi 1) = w1 < < − I 1 − I { 23··· } O { ··· w# i(Gi− ) . Draw the bipartite tree Ti in i(Gi− ), i(Gi− ) with the left vertex set i(Gi 1) = O 1 } TI 1 O 1 I − v1 < < v# i(Gi− ) so that the vertices v1 < < v# i(Gi− ) are ordered top to bottom { ··· I 1 } ··· I 1 vertically on the left. Similarly, the right vertices w1 < < w# i(Gi− ) are drawn top to bottom ··· O 1 vertically on the right. We let the degree of vertex wj in Ti be cj +1. The above uniquely determines the noncrossing bipartite tree Ti on left and right vertex sets i(Gi 1) and i(Gi 1). With tree Ti I − O − constructed we do a reduction at vertex i to obtain G := (G )(i) with an inheritance framing. i i 1 Ti We iterate this for i = 1, . . . , n 1. See Figure 14 for an example.− − Thus, from the integer flow ifl( ) we obtain a tuple of bipartite noncrossing trees (T2,T3,...,Tn 1) (i) · − such that Gi := (Gi 1) for i = 2, . . . , n 1 and G1 := G. Since Gn 1 has no incoming or outgoing − Ti − − edges to vertices i = 2, . . . , n 1 then Gn 1 consists of two vertices 1 and n and #E(G) n + 2 multiple edges. Thus is− a (#(G) n− + 1)-simplex. − FGn−1 − Recall that each such multiple edge e in Gn−1 is a sum of edges of the original graph of G as explained in the beginning of this section. SuchF sum of edges corresponds to a route in the graph G, i.e. a directed path from vertex 1 and n in G. We denote the unit flow in G along the route corresponding to edge e in Gn 1 by ρ(e) and we let − (G, ) ∆ ≺ := ConvHull ρ(e) e E(Gn 1) ifl { | ∈ − } (G, ) be the simplex with vertices ρ(e). Note that ∆ ≺ is integrally equivalent to , and it is a ifl FGn−1 subset of G. PostnikovF and Stanley proved Theorem 3.4 by showing that this iterative construction of sim- int plices from integer flows yields a triangulation of G. Denote by G (0, d2, . . . , dn 1, i di) the F F − − set of nonnegative integer flows on the graph G with netflow (0, d2, . . . , dn 1, i di). − −P Theorem 7.2 (cf. [16, 6.1]). Given a framed graph (G, ), the set of simplices P § ≺ (G, ) int ∆ ≺ ifl G (0, d2, . . . , dn 1, di) , { ifl | ∈ F − − } Xi where d = indeg (G) 1, are the top simplices of a unimodular triangulation of . i i − FG Remark 7.3. Note that the triangulation in [16, 6.1] comes from the top to bottom framing of the graph. Theorem 7.2 yields a triangulation for any§ framing of the graph. The proof in [16, 6.1] adapts readily for an arbitrary framing. Indeed, more general triangulations can be constructed§ in the above way that do not depend on a fixed framing of the graph G; we only need to specify some (any) ordering of edges at each vertex as we do the reductions.

7.2. The set of DKK triangulations equals the set of framed PS triangulations. In this section we show that with a fixed framing (G, ) the DKK triangulations and the PS triangulation are identical. In effect, the set of DKK triangulations≺ equals the set of framed PS triangulations. We also give an explicit bijection between the objects indexing a DKK triangulation of G for a framing of G and a framed PS triangulation of , namely a bijection between maximal cliquesF of G with FG respect to a fixed framing and nonnegative integer flows of G with netflow (0, d2, . . . , dn 1, i di). (G, ) − − The following results show that the vertices of a simplex ∆ ≺ correspond to a maximal clique ifl P (G, ) of the framed graph (G, ). Recall that the simplex ∆ ≺ is integrally equivalent to the flow ≺ ifl polytope Gn−1 of a graph Gn 1 consisting of vertices 1 and n and #E(G) n + 2 multiple edges F − (G, ) − (1, n) and that the set of simplices ∆ ≺ , as ifl runs over all flows in int(0, d , d , d , d , d ) ifl FG 2 3 4 5 − i i forms the top dimensional simplices of a unimodular triangulation of G as shown in Theorem 7.2. F P Proposition 7.4. Given a framed graph (G, ) with vertices [n] and a nonnegative integer flow int ≺ ifl G (0, d2, d3, d4, d5, i di), where di = indegi(G) 1, the routes of G along which the unit ∈ F − 24 − P e13 e24 1 F F e12 1 G G2 Graph G with 1 1 1 framing in red. 2 2 2 2 e12 e23 e34 1 2 3 4 F 0 G2

reduction at vertex 2 reduction at vertex 2 1 e12 + e24 1 e12 + e24 0 1 1 0 e12 + e23 1 2 e12 + e24 2 e12 + e23 partial nonnegative flow 2 partial nonnegative flow e12 + e23 with netflow (0, 1, ·, ·) with netflow (0, 1, ·, ·) obtained from noncrossing tree obtained from noncrossing tree 1 e1 + e e12 + e24 1 12 24 e12 + e23 2 e12 + e24 e e13 13

1 2 3 e34 4 1 2 3 e34 4 e + e23 e12 + e23 12

reduction at vertex 3 reduction at vertex 3

e13 + e34 0 e13 + e34 1 1 e12 + e23 + e34 1 2 0 1 2 e + e23 + e34 2 12 e + e23 + e34 nonnegative flow obtained 12 nonnegative flow obtained with netflow (0, 1, 1, −2) with netflow (0, 1, 1, −2) from the noncrossing trees from the noncrossing trees

1 1 e12 + e24 e12 + e24 2 e12 + e24 e13 + e34 0 graph G2 graph G 1 e13 + e34 2 e12 + e23 + e34 2 2 1 e12 + e23 + e34 4 1 e12 + e23 + e34 4

Figure 14. Reductions executed at vertex 2 and 3 of the framed graph G. Non- crossing trees encoding the reduction are displayed with all edges labeled. The nonnegative flow on G with netflow (0, 1, 1, 2) is built. The flow polytope is − FG dissected into two simplices corresponding to G2 and G20 .

(G, ) flows give the vertices of the simplex ∆ifl ≺ form a maximal clique with respect to the coherence relation in (G, ). ≺ (G, ) int Proof. Recall that ∆ ≺ Gn− , for some Gn 1 as described in Section 7.1. Recall that a ifl ≡ F 1 − sequence of graphs G1 := G, G2,...,Gn 1 encode the successive reductions leading to the simplex − (G, ) int ≺ ∆ifl Gn−1 . Graph G has a framing, and the framing of graph Gi, i [2, n 1], is the ≡ F 25 ∈ − (1, 0, 2) T3 T2 (4)

(2) (3) G3 = (GT )T (2) 2 3 G G1 G2 = G ifl T2 1 0 3 2 2 4 −5 1 0 0 −1 1 0 −1 1 −1

Figure 15. Example in the Postnikov–Stanley triangulation of of how to FG find a simpex G3 from an integer flow ifl in G(0, d2, d3, d2 d3) where di = indeg (G) 1.F Each step of the subdivision isF encoded by− noncrossing− trees T i − i+1 that are equivalent to compositions (b1, . . . , br) of # i+1(Gi) 1 with # i+1(Gi) parts. These trees or compositions are read from the integerI flow.− The framingO used is top to bottom.

inheritance framing obtained from the framing of Gi 1. Suppose that to the contrary, there are (G, ) − two vertices of the simplex ∆ifl ≺ , which correspond to non-coherent routes P and Q in G. Suppose that P and Q are not coherent at the common inner vertex v. Suppose that the smallest vertex after which P v and Qv agree is w1 and the largest vertex before which vP and vQ agree is w2. Let 1 1 the edges incoming to w1 be eP and eQ for P and Q, respectively, and let the edges outgoing from 2 2 w2 be eP and eQ for P and Q, respectively. Since P and Q are not coherent at v, this implies that either e1 e1 and e2 e2 or e1 e1 and e2 e2 . We also have that P ≺in(w1) Q Q ≺out(w2) P Q ≺in(w1) P P ≺out(w2) Q the segments of P and Q between w1 and w2 coincide. 1 Denote by p the sum of edges between w1 and w2 on P . Denote by (eZ + p), for Z P,Q , 1 ∗ ∈ { } the sum of edges left of w2 that are edges in Z (including eZ in particular). After a certain number of reductions executed according to the framing, we are about to perform the reduction at vertex w2. This reduction involves deleting w2 and the edges incident to it, and adding the edges obtained from the noncrossing tree T we constructed based on the ordering of the incoming and outgoing edges at w2. In such a noncrossing tree, the vertex corresponding to the edge stemming from (e1 + p), Z P,Q , is above the vertex (e1 + p), where Z is the complement of Z in P,Q , in ∗ Z ∈ { } ∗ Z { } the left partition of the vertices of T . On the other hand, the vertex corresponding to e2 is above Z 2 the vertex corresponding to eZ in the right partition of the vertices of T . Thus, it is impossible to obtain both routes P and Q as vertices of since that would force connecting (e1 + p) FGn−1 ∗ Z and e2 as well as (e1 + p) and e2 in T . This would make a crossing in the noncrossing tree T , a Z ∗ Z Z contradiction.  Proposition 7.4 justifies the following definition. Definition 7.5. Given a framed graph (G, ) on the vertex set [n], let ≺ (G, ) int max Λ ≺ : G (0, d2, . . . , dn 1, di) (G, ) F − → C ≺ Xi (G, ) (G, ) be the map defined by Λ ≺ (ifl) = C, where the vertices of the simplex ∆ifl ≺ are the unit flows along routes in the maximal clique C and d = indeg (G) 1. i i − (G, ) Example 7.6. Figure 17 gives an example of the bijection Λ ≺ between the two integer flows int in G (0, d2, d3, d2 d3) and the two maximal cliques with respect to the framing of G given in FigureF 14. − − 26 ifl 0 Λ 0 2

1 1 4 1 2 3 −6 15 T5 T2 26 T3 14 T4 12 + 25 25 13 36 12 + 24 46 13 + 35 56 12 24 35 13 + 34 13 + 34 + 45 23 12 + 23 34 12 + 23 + 34 45 12 + 23 + 34 + 45

(G, ) Figure 16. Example of the bijection Λ = Λ ≺ between an nonnegative integer flow ifl in (0, d , d , d , d , d ) where G = K and d = indeg (G) 1. Below FG 2 3 4 5 − i i 6 i i − are the noncrossing trees Ti with left vertices i(Gi 1) and right vertices i(Gi 1) written as sums of edges of G (ijPis shorthand forI the− edge (i, j)). The framingO used− is top to bottom.

integral flow outcome graph maximal clique

1 Λ e12 + e24

1 e + e24 12 e13 + e34 G G2 e13 + e34 1 1 e12 + e23 + e34 0 e12 + e23 + e34 1 2 2 1 2 4 e12 + e23 + e34 e12 + e23 + e34

Λ 1 e12 + e24 1 e12 + e24 0 2 2 G e + e24 e12 + e24 G 2 12 e + e 1 13 34 e + e 0 1 13 34 2 1 e12 + e23 + e34 4 2 e12 + e23 + e34

Figure 17. The graphs G, G2 and G20 as well as the edge labels eij are of the edges of the graph G are as in Figure 14. The four paths on the top correspond to the vertices of the simplex given by G2. The four paths on the bottom correspond to the vertices of the simplex given by G20 . Both sets of paths are coherent in the top to bottom framing of G given in Figure 14.

(G, ) Example 7.7. Figure 16 gives a larger example of the bijection Λ ≺ between an integer flow in int (0, 0, 1, 2, 3) and a maximal clique of K6. FK6 − We now have: Theorem 7.8. Given a framed graph (G, ) on the vertex set [n] the Danilov-Karzanov-Koshevoy ≺ triangulations of G with respect to this framing is the framed Postnikov-Stanley triangulations of F (G, ) G with respect to the same framing. Moreover, the map Λ ≺ defined above is a bijection between F 27 int nonnegative integer flows in G (0, d2, . . . , dn 1, i di), where di = indegi(G) 1, and maximal cliques in max(G, ). F − − − C ≺ P Proof. Fix a framing (G, ). Proposition 7.4 shows that the framed PS triangulation with respect to this framing is the same≺ as the DKK triangulation with respect to this framing. Therefore, any (G, ) DKK triangulation is a framed PS triangulation. In particular, Λ ≺ is a bijection that simply sends one set of labelings of a fixed triangulation of G into another set of labelings of the very same triangulation of . F FG  (G, ) We conclude by noting that there is a nice way to describe the inverse of the map Λ ≺ : int Lemma 7.9. Fix a framed graph (G, ) and a flow ifl G (0, d2, . . . , dn 1, i di), where ≺ ∈ F − − d = indeg (G) 1. If Λ(G, )(ifl) = C, then each edge e of the graph G appears ifl(e) + 1 times as i i ≺ P an edge of one of− the paths ending in v = fin(e) in the set (not multiset!) P v P C, v = fin(e) of prefixes of routes in the clique C. In particular, given a maximal clique{ C| ∈ max(G, ) the} (G, ) 1 ∈ C ≺ inverse (Λ ≺ )− (C) is given by (G, ) 1 ((Λ ≺ )− (C))(e) = n(e) 1, − where n(e) is the number of times edge e appears in set of prefixes P v P C, v = fin(e) . { | ∈ } (G, ) Proof. By the construction of Λ ≺ , from the integer flow ifl( ) we obtain a tuple of noncrossing · (i) bipartite trees (T2,T3,...,Tn 1) such that G1 = G and Gi := (Gi 1) for i = 2, . . . , n 1 where − − Ti − Gn 1 is a graph with vertices 1 and n and #E(G) n + 2 multiple edges where each multiple edge is a− sum of edges of the original graph G defining− a route of the maximal clique C. The edges of intermediate graphs G2,...,Gn 2 for i = 2, . . . , n 2 encode prefixes of the routes in − − the clique C as follows: for the edge e = (i, j) in i(G) = i(Gi 1), the tree Ti in i(Gi− ), i(Gi− ) O O − TI 1 O 1 has ifl(e) + 1 tree-edges incident to e by definition. Therefore, the edge e appears exactly ifl(e) + 1 times in the set of prefixes of the routes P v P C, v = fin(e) . The statement about (G, ) 1 { | ∈ } (Λ ≺ )− (C) then follows readily.  Example 7.10. We continue with Example 7.7 illustrated in Figure 16. Edge e = (3, 4) has flow ifl(e) = 1 and there are two paths ending in vertex 4 containing e in the corresponding clique (G, ) C := Λ ≺ (ifl), namely the paths consisting of edges (1, 2), (2, 3), (3, 4) and of edges (1, 3), (3, 4). Note that the path (1, 3), (3, 4) is the prefix of two routes in the clique, however, we count it here just once since in Lemma 7.9 we are looking at the set of prefixes of the routes in the clique and not a multiset of prefixes.

Acknowledgments The authors are grateful to Alexander Postnikov for generously sharing his insights and ques- tions. The authors are also grateful to the anonymus referee for numerous helpful comments and suggestions. AHM and JS would like to thank ICERM and the organizers of its Spring 2013 pro- gram in Automorphic Forms during which part of this work was done. The authors also thank the SageMath community [28] for developing and sharing their code by which some of this research was conducted.

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Department of Mathematics, Cornell University, Ithaca, NY 14853 and School of Mathematics, Institute for Advanced Study, Princeton, NJ 08540

Department of Mathematics and Statistics, UMass, Amherst, MA, 01003

Department of Mathematics, North Dakota State University, Fargo, ND 58102 E-mail address: [email protected], [email protected], [email protected]

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